Optical fibre based artificial compound eyes for direct static imaging and ultrafast motion detection

Natural selection has driven arthropods to evolve fantastic natural compound eyes (NCEs) with a unique anatomical structure, providing a promising blueprint for artificial compound eyes (ACEs) to achieve static and dynamic perceptions in complex environments. Specifically, each NCE utilises an array of ommatidia, the imaging units, distributed on a curved surface to enable abundant merits. This has inspired the development of many ACEs using various microlens arrays, but the reported ACEs have limited performances in static imaging and motion detection. Particularly, it is challenging to mimic the apposition modality to effectively transmit light rays collected by many microlenses on a curved surface to a flat imaging sensor chip while preserving their spatial relationships without interference. In this study, we integrate 271 lensed polymer optical fibres into a dome-like structure to faithfully mimic the structure of NCE. Our ACE has several parameters comparable to the NCEs: 271 ommatidia versus 272 for bark beetles, and 180o field of view (FOV) versus 150–180o FOV for most arthropods. In addition, our ACE outperforms the typical NCEs by ~100 times in dynamic response: 31.3 kHz versus 205 Hz for Glossina morsitans. Compared with other reported ACEs, our ACE enables real-time, 180o panoramic direct imaging and depth estimation within its nearly infinite depth of field. Moreover, our ACE can respond to an angular motion up to 5.6×106 deg/s with the ability to identify translation and rotation, making it suitable for applications to capture high-speed objects, such as surveillance, unmanned aerial/ground vehicles, and virtual reality.

Due to the diffraction and the nonideal properties of lenses [1][2] , the intensity distribution h(x, y) of the blurred circle follows a two-dimensional Gaussian distribution, where σ is the spread parameter, which has a linear relationship with d2 (k is a constant): 2 , here 0 In fact, h(x, y) is the point spread function of this camera, as discussed below.We can combine Eqs.(S1) and (S3) to formulate the relationship between the spread parameter σ and the object distance u: For the camera used in our setup, the variables k, D, s and f are fixed camera parameters.
Thus, Eq. (S4) can be simplified as where m and c are constants.
To determine the values of m and c using linear fitting, the values of σ and u under several different conditions should be measured.We first consider how to measure σ.
We introduce a step edge function along the y direction in the image plane f(x, y), which can be formulated as where a is the initial intensity, b is the height of the step, and u(x) is the standard unit step function.The observed image g(x, y) is then the convolution of f(x, y) and the point spread function h(x, y): where * represents the convolution operation.The derivative of g along the gradient direction can be written as where δ(x) is the derivative of u(x) along the x direction, which has the form of a Dirac delta function.This expression can be written as , which is also called the line spread function, can be used to represent this line integral: Then, Eq. (S9) can be written as The integration of this equation along the The integral of the point spread function should be unity, i.e., ( , ) 1 h x y dxdy Therefore, we have Thus, θ2(x) can be expressed as 2 ( ) Since the point spread parameter σ of a line spread function 3 is the standard deviation of the line spread function θ2(x), we have The combination of Eqs.(S18) and (S19) yields The above expression can be rewritten as Here, we use some variables to simplify the equation, , , , Then, the equation can be written in matrix form as follows: As there are two variants, u2 and v2, a single point is insufficient for calculating their values.To address this issue, a calculation pixel window with a size of w  w is established.
To calculate the variants, we assume that the pixels in the same window follow the same motion.Consequently, each window contains w 2 pixels, resulting in w 2 functions.The equation can then be expressed in matrix form as follows: . .This is an overdetermined linear equation, and the least square method is adopted as follows: With this approach, the optical flow u2 and v2 in a window can be computed, and after calculating the values for all windows in an image, the overall optical flow can finally be obtained.

Figure S2 |
Figure S2 | Images of the laser spots projected onto the full-field camera.The images are

Figure S3 |
Figure S3 | Static imaging results of different object patterns captured by the ACEcam.

Figure S4 |
Figure S4 | Setups for optical flow detection.a, The ACEcam is positioned 10 mm in front

Figure S5 |
Figure S5 | Experimental setup to generate high angular velocities for the dynamic response

Figure S6 |
Figure S6 | Dynamic angular perception results when fflicker = 24 Hz.The lower half shows

Figure S7 |
Figure S7 | Response characteristics of spiking neurons (a) and nonspiking graded

Figure S8 |
Figure S8 | Schematic diagram of the angle of the tangent line.The Cartesian coordinate

Figure S9 |
Figure S9 | Acceptance angles of the spherical microlens optical fibres for the microlenses with different radii.a, The upper limit angle, which is theoretically calculated, versus the radial position at which the light hits the microlens surface (Fig. 6c in the main

Figure S10 |
Figure S10 | Divergence angles of the optical fibres capped with conical microlenses with different half-apex angles.Red paths represent light reflected from the upper core/cladding interface of the optical fibre, green paths represent light reflected from the lower core/cladding interface of the optical fibre, and blue paths represent light emitted from the rounded tip of the optical fibre.a, When the cone has a half-apex angle   43 o , the rays from the upper core/cladding interface go out at a downward angle, and those from the lower core/cladding interface go out at an upward angle, forming a solid circle on the observation screen.The acceptance angle is determined by the upper limit angle αupper.b, When 31 o <  < 43 o , the rays from the upper core/cladding interface and the lower core/cladding interface both travel downwards, causing a hollow central region on the observation screen.The rays from the hollow central region cannot be collected by the conical microlens optical fibre.c, The tip of the cone is rounded to prevent the appearance of the hollow central region.

Figure S11 |
Figure S11 | Light paths in deviated conical microlens optical fibres for the alignment

Figure S12 |
Figure S12 | Imaging property of a lens using geometrical optics.Here, p is the image where x represents the average of the line spread function.Thus, it follows that of an observed image is obtained, the line spread function θ2(x) can be determined.Then, the average x and the point spread parameter σ can be obtained.Therefore, the point spread parameter σ indirectly represents the gradient at the edge.If several values of the point spread parameter σ and the object distance u are given, the values of m and c can be determined using Eq.(S5).More importantly, m is the slope of the relationship between σ and u -1 and thus represents the extent to which a camera ' s imaging quality (characterized by the gradient at the edge) is affected by a change in the object distance.In this work, this parameter m is defined innovatively as the critical parameter.